Conformal symmetry and the
Virasoro algebra
THESIS
submitted in partial fulfillment of the requirements for the degree of
BACHELOR OF SCIENCE in
PHYSICS
Author : B.W. van Kootwijk
Student ID : s1852337
Supervisors : Prof.dr. K.E. Schalm, Dr. R.I. van der Veen Leiden, The Netherlands, July 11, 2020
Conformal symmetry and the
Virasoro algebra
B.W. van Kootwijk
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
July 11, 2020
Abstract
In this thesis we study conformal invariance of two-dimensional physical systems, and its connections to the Virasoro algebra. We first discuss how
symmetries relate to Lie algebras, and how the Virasoro algebra corresponds to conformal symmetries. We then study the theory of the free open bosonic string. Starting with the Nambu-Goto action, we first solve the equations of motion of the classical string using the light-cone
gauge. From there, we construct the quantum theory and show it is indeed a representation of the Virasoro algebra. We then look more directly into representations of the Virasoro algebra. We define lowest weight representations and Verma modules, and discuss reducibility of
Introduction . . . . 7
1 Preliminaries . . . . 9
1.1 Metrics . . . 9
1.1.1 Distances . . . 9
1.1.2 Angles and conformal maps . . . 12
1.2 Lagrangian mechanics . . . 15
1.2.1 The Lagrangian . . . 15
1.2.2 The free relativistic point particle . . . 17
1.3 The quantum formalism . . . 19
1.3.1 The quantum harmonic oscillator . . . 21
1.3.2 Symmetries and Noether’s theorem . . . 23
2 The Virasoro algebra . . . . 25
2.1 Lie algebras . . . 25
2.2 Lie algebra representations and the universal enveloping al-gebra . . . 28
2.3 The Witt and Virasoro algebras . . . 35
3 Free bosonic string theory . . . . 41
3.1 The free string . . . 41
3.1.1 Useful parameterisations and solving the (classical) equations of motion. . . 43
3.1.2 Solving the equations . . . 46
3.1.3 The light-cone gauge . . . 49
3.2 Quantisation of the string . . . 50
3.2.1 Commutation relations and normal ordering . . . 53
3.3 Lorentz symmetry and conserved quantities. . . 57
3.4 Conformal properties of the world sheet . . . 59
4 Representations of the Virasoro algebra . . . . 61
CONTENTS 5
4.2 Finding singular vectors . . . 67
4.2.1 Unitarity . . . 74
Conclusion. . . . 77
Introduction
Many algebraic structures in theoretical physics arise through symme-tries of physical systems. Symmetry is a fundamental concept in modern physics, and it is no wonder Nobel laureate P. W. Anderson once said: “it is only slightly overstating the case to say that physics is the study of symmetry” [And72]. The essential role of symmetry becomes clear via Noether’s the-orem, which states that families of spacetime symmetries correspond to conserved quantities. For example, invariance under translations induces conservation of momentum, and similarly rotational symmetry results in conserved angular momentum. In these two cases, the set of symmetries forms an algebraic group under composition, since both translations and rotations are invertible. However, one often encounters symmetries which do not follow this pattern: these symmetries may not be invertible or may only be defined locally. A set of such symmetries does not generally form a group, yet one can still study them by looking at their infinitesimal genera-tors. These generators naturally form a Lie algebra, a vector space endowed with a certain antisymmetric bilinear operation.
Apart from translating or rotating the target space, another natural trans-formation is scaling. Scale invariance might at first glance seem an un-likely property of any physical system, as physical systems usually have some defined notions of size and mass. However, certain theories such as the Ising model do, under certain conditions, show scale invariance. A generalisation of proper scaling transformations are transformations which infinitesimally only scale, but globally may exhibit more involved behaviour. Such transformations are exactly those transformations that leave angles invariant, which are referred to as conformal transformations. Generally we may take the target space to be Rn for some n ∈ Z≥2. For
de-fined, i.e. it is defined on some open subset of Rn, can be extended to a globally defined invertible conformal transformation [Sch08]. This results in a finite number infinitesimal generators. In the two-dimensional case, however, such extensions are not possible in general, and one obtains an infinite set of generators. Conformal invariance in two dimensions is there-fore very restrictive compared to higher dimensional cases, which makes it easier to determine the dynamics of conformally invariant systems in two dimensions. In this thesis, we solely focus on two-dimensional conformal theories and the Lie algebra structures they contain, namely the Witt and Virasoro Lie algebras. In particular, we investigate the algebraic structure of the simplest of such theories which is given by lowest weight representa-tions of the Virasoro algebra. First, however, we will discuss one particular physical example of a conformal theory, given by bosonic string theory. String theory is often advocated as a theory of everything. In this theory, the fundamental building blocks of nature are thought to be strings in-stead of elementary particles. While the theory was originally meant to merely describe the strong interaction, it was later found to contain parti-cles with the properties of the photon and the hypothetical graviton, the mediator of gravity [Ric16]. This led to string theory being considered in a much broader context, namely as a theory in which both quantum mechanics and Einsteins theory of relativity are united. Although a multi-tude of string theories has risen over the years, for our purposes it suffices to study the simpler bosonic string theory, and leave out the supersym-metrical counterparts involving fermions. The strings in string theory are micoroscopic one-dimensional objects which trace out a two-dimensional surface in spacetime as they propagate through time. This surface is called the world sheet. Conformal transformations of the world sheet leave the dy-namics of string invariant, which indeed makes string theory a conformal field theory in two dimensions. In the quantum theory of the string one therefore obtains a representation of the Virasoro Lie algebra, as we will show for the open bosonic string.
Inchapter 1 we cover the necessary groundwork on which the following chapters are build. This includes a definition of metrics, the framework of Lagrangian mechanics and a short introduction into quantum mechanics. Inchapter 2we develop the basics of the theory of Lie algebras and their representations, and investigate the Lie algebras corresponding to confor-mal transformations. Inchapter 3we discuss the open bosonic string and formulate a quantum theory. We finish inchapter 4by studying elemen-tary representations of the Virasoro algebra, namely lowest weight repre-sentations.
Chapter
1
Preliminaries
In this chapter we give a short review of general concepts that will be used and discussed within this thesis. These concepts include metrics on vector spaces, Lagrangian mechanics and quantum theory.
1.1
Metrics
Throughout this thesis we will often encounter notions of length, area and angles on different sets. Although the reader should be familiar with these concepts, they might be surprised as we use a slightly generalised notion of length and area, which might result in negative values. We will there-fore introduce precise definitions for these concepts.
1.1.1
Distances
Metrics are defined through inner products, for which we use a more gen-eral definition than is conventional.
Definition 1.1. Let V be a real vector space. An inner product on V is a map V×V →R,(v, w) 7→ v·w which is bilinear, symmetric, and admits a basis B of V such that for all b, c ∈ B we have
hb, ci = (
0 if b6=c,
±1 if b=c. (1.1)
For finite dimensional vector spaces this last property simple means the inner product is non-degenerate, since it directly implies no nonzero
vec-tor is orthogonal to every basis vecvec-tor. One should note we explicitly do not require an inner product to be positive definite, which is usually in-cluded in the definition. As a consequence, the inner product of a vector with itself might be zero or negative. The advantage of this is that if we want to view our vector space as spacetime, we can distinguish between time and space directions. Any vector whose inner product with itself is negative can be thought of as pointing in a timelike direction. In line with this idea, a vector v ∈V with v2 >0, v2=0, v2 <0 will be called spacelike, lightlike or timelike respectively. We can now use inner products to define metrics.
Definition 1.2. Let n ∈ Z≥1, let M be an n-dimensional real smooth manifold, and for each x ∈ M let TxM be its tangent space. A metric on M is a map
g : M → Bil(TxM×TxM,R) such that for all x ∈ M the map g(x) is a real
inner product and such that for any smooth curve γ : (−1, 1) → M the map t7→ g(γ(t))(γ0(t), γ0(t))is smooth.
A manifold with such a metric is called a pseudo-Riemannian manifold. Given two vectors v, w in some tangent space TxM we will often shorten
notation and write v·w instead of g(x)(v, w), if it is clear what is meant by this. Accordingly, v2 means g(x)(v, v). A simple but very important example of a pseudo-Riemannian manifold is Minkowski space.
Definition 1.3. For d ∈ N, the d+1-dimensional Minkowski space is the vector spaceR(1,d) := R1×Rd. We write its elements as x = (x0,~x)for x0 ∈
R1 and~x = (x1, . . . , xd) ∈ Rd. This vector space is endowed with the dot
productR(1,d)×R(1,d) →R given by x·y= (x0,~x) · (y0,~y) := −x0y0+ d
∑
i=1 xiyi. (1.2) Here we identifiedR(1,d)with its own tangent space.A common alternative for the definition of this inner product is via matrix multiplication. In order to do this, one defines the(d+1) × (d+1)matrix
η = −1 0 0 . . . 0 0 1 0 . . . 0 0 0 1 . . . 0 .. . ... ... . .. ... 0 0 0 . . . 1 , (1.3) and x·y=y|ηx, (1.4)
1.1 Metrics 11
where all multiplication on the right-hand side is ordinary matrix multi-plication. Yet another way of writing such an inner product is via Einstein notation. Within this notation, summation symbols are suppressed and instead implied by repeated higher and lower indices. For example
ηµνxµyν := d
∑
µ=0 d∑
ν=0 ηµνxµyν =y|ηx=x·y. (1.5)Any linear map L : R(1,d) → R(1,d)
which preserves this inner product is called a Lorentz transformation. In particular such transformations preserve the “distance squared”. Special relativity tells us that any physical system is invariant under a Lorentz transformation; the laws of physics do not change when a Lorentz transformation is applied.
A pseudo-Riemannian manifold has more structure than a general man-ifold, as it provides a notion of distance and angles on the manifold. One should here distinguish between curves that follow spacelike trajec-tories and ones that follow timelike trajectrajec-tories. That is, given a curve
γ : [0, 1] → M such that γ0 is always spacelike, the length of γ is defined
as length(γ):= Z 1 x=0dx q γ0(x)2, (1.6)
whereas for such γ where γ0 is always timelike, the proper time elapsed along γ is proper time(γ):= Z 1 t=0dt q −γ0(t)2. (1.7)
Proper time is an important concept, as it can be used to determine the behaviour of a single relativistic point particle. However, it will not be sufficient in string theory, for the simple reason a string traces out a two-dimensional surface, which has both space and time directions. In this specific case, we will be faced with a map of the type γ : [0, 1]2 → M, (σ, τ) 7→ γ(σ, τ)such that ∂γ∂σ is spacelike and ∂γ∂τ is lightlike, and we need
to define its area. This definition would have to depend on ∂γ∂σ and ∂γ∂τ and be invariant under reparameterisations. The correct definition in this case is area(γ) = Z 1 τ=0 dτ Z 1 σ=0 dσ s ∂γ ∂τ · ∂γ ∂σ 2 − ∂γ ∂τ 2 ∂γ ∂σ 2 . (1.8)
Since all of ∂γ∂τ · ∂γ ∂σ 2 , −∂γ ∂τ 2
, and ∂γ∂σ2 are positive, the quantity in the square root is positive, so this expression is well-defined. Moreover, it depends, up to a sign, linearly on both derivatives, which is an expected property of any area functional.
1.1.2
Angles and conformal maps
Returning to the setting introduced in the previous section, where we have some manifold M with a metric, the metric gives rise to another interesting notion, namely that of angles. However, we will only be using angles in the case of a positive definite metric, so this will be assumed in the remaining of this chapter. Angles are defined in the following manner. Definition 1.4. Given two curves γ, δ : (−1, 1) → M intersecting at some x ∈ M at time t = 0 (by which we mean that γ(0) = δ(0) = x), the angle α
between γ and δ is defined by
cos(α) := g(x)(γ 0(0), δ0(0)) p g(x)(γ0(0), γ0(0) ·g(x)(δ0(0), δ0(0) (1.9) = γ 0(0) · δ0(0) p γ0(0)2δ0(0)2 . (1.10)
This expression should be familiar, as it is simply a generalisation of the formula x·y = cos(α)kxk kyk which holds in Euclidean space. We can
now define so called conformal maps, which are maps between manifolds that preserve angles.
Definition 1.5. Let n ∈ Z≥1, let M and M0 be n-manifolds and let g and g0 be metrics on M and M0 respectively. A conformal map from M to M0 is a differentiable map f : M → M0such that for all x ∈ M and for all v, w ∈ TxM
we have
g0 f(x)
f0(x)v, f0(x)w
=λ2(x) ·g(x)(v, w). (1.11)
for some smooth map λ : M→R>0.
It should be clear that a conformal map preserves angles as defined in (1.9). As we are specifically interested in conformal maps and conformal invariance in two dimensions, we will restrict ourselves to the case M = R2. Here R2 is equipped with the standard metric g(x) = h·,·i, which
assigns to every point the standard (Euclidean) inner product. Moreover, in order to classify the conformal maps onR2, we use the isometryC −→∼
1.1 Metrics 13
R2 given by a+bi 7→ (a, b) to identifyR2with the complex plane. Since
we are working with a constant metric, (1.11) simplifies considerably to f0(
x)v, f0(x)w =λ2(x) · hv, wi. (1.12)
The set of all conformal maps is now easily determined; we show all con-formal maps on the plane are either holomorphic or antiholomorphic. Proposition 1.6. Let U ⊆R2be an open non-empty subset and let f : U →R2
given by z 7→ (f1(z), f2(z)) be a conformal map (where both U and R2 are
equipped with the standard metric mentioned above). Then, if f is viewed as a complex function, f is either holomorphic or antiholomorphic. In other words, either f or its complex conjugate is holomorphic.
Proof. First note that for all z ∈ U the derivative of f is just its Jacobian matrix d f(z) = ∂xf1 ∂yf1 ∂xf2 ∂yf2 , (1.13)
where ∂xfiand ∂yfi are the partial derivatives of fito the first respectively
the second coordinate. By substituting the (orthonormal) standard basis vectors in the place of v and w in (1.11), the right-hand simplifies to either zero or λ2(x). To be more concrete, let i, j ∈ {1, 2}, and let v = ei and
w =ej. Then by (1.11) we get λ2(x) ·δi,j =λ2(x) · hv, wi = ∂xf1 ∂yf1 ∂xf2 ∂yf2 ei, ∂xf1 ∂yf1 ∂xf2 ∂yf2 ej = ∂if1 ∂if2 , ∂jf1 ∂jf2 =∂if1·∂jf1+∂if2·∂jf2, (1.14) and in particular (∂xf1)2+ (∂xf2)2 =λ2(x) = (∂yf1)2+ (∂yf2)2 (1.15) and ∂xf1·∂yf1+∂yf2·∂xf2 =0. (1.16)
To prove f is either holomorphic or antiholomorphic, we now transition explicitly to the complex plane. We write z = x+iy and ¯z = x−iy (and equivalently x = z+2¯z and y = z−2i¯z), and we view f as a complex valued function via f = f1+i f2. We will finish the proof by showing that either
∂zf =0, or ∂¯zf = 0, since these conditions are equivalent to f being
holo-morphic in the first case, and antiholoholo-morphic in the second case. To do this, we will first need explicit expressions for ∂zand ∂¯z.
For a general function g(x, y)we may apply the chain rule to find explicit expressions for ∂z = ∂z∂ and ∂¯z= ∂ ¯∂z in terms of ∂xand ∂y, namely as
∂zg(x, y) = ∂g ∂x ∂x ∂z + ∂g ∂y ∂y ∂z = ∂g ∂x 1 2+ ∂g ∂y 1 2i = 1 2(∂x−i∂y)g (1.17) and ∂¯zg(x, y) = ∂g ∂x ∂x ∂ ¯z + ∂g ∂y ∂y ∂ ¯z = ∂g ∂x 1 2− ∂g ∂y 1 2i = 1 2(∂x+i∂y)g. (1.18) Hence we conclude ∂z = ∂x −i∂y 2 and ∂¯z = ∂x+i∂y
2 . We can now directly
calculate the product(∂zf)(∂¯zf) by splitting the real and imaginary part.
We have 4(∂zf)(∂¯zf) = (∂x−i∂y)(f1+i f2)· (∂x+i∂y)(f1+i f2), so∗ 4 Re[(∂zf)(∂¯zf)] = (∂xf1+∂yf2)(∂xf1−∂yf2) + (∂xf2−∂yf1)(∂xf2+∂yf1) = (∂xf1)2− (∂yf2)2+ (∂xf2)2− (∂yf1)2 =0 and 4 Im[(∂zf)(∂¯zf)] = (∂xf2−∂yf1)(∂xf1−∂yf2) − (∂xf1+∂yf2)(∂xf2+∂yf1) = −2∂xf1∂yf1−2∂xf2∂yf2 =0.
Both the “= 0” follow directly from (1.15). Hence either ∂zf =0 or ∂¯zf =
0, which completes the proof. Note that in the case where in some point both ∂zf and ∂¯z are zero, we simply have f0 = 0 which implies λ = 0,
which is excluded by definition of conformal maps.
Remarkably, the situation in two dimensions is unique, in the sense that the set of conformal transformations is “very large”. It can be shown that
∗Here we use the identities Re(zw) = Re(z)Re(w) +Im(z)Im(w) and Im(zw) =
1.2 Lagrangian mechanics 15
in higher dimensions, all conformal transformations can be obtained by taking combinations of translations, rotations, dilatation, reflections, and so called special conformal transformations [Sch08]. The latter are given by
x 7→ x− hx, xib
1−2hx, bi + hx, xihb, bi
for some b ∈ Rn, and are obtained by doing an inversion (x 7→ hx,xx i), translating by b and doing another inversion. The set of conformal trans-formations (in more than two dimensions) is therefore generated by only a finite set of infinitesimal transformations [Sch08]. More formally, the Lie algebra corresponding to these transformations is finite dimensional. The contrary is true in two dimensions, as the set of holomorphic trans-formations requires an infinitely set of generators. This implies conformal invariance in two dimensions is much more restrictive than it would be in higher dimensions, which motivates the study of this specific case.
1.2
Lagrangian mechanics
As we will see, string theory provides an example of conformal invariance. The theory of the open bosonic string inchapter 3will be formulated via a Lagrangian. We will give a brief introduction to Lagrangian mechanics and treat the example of the relativistic point particle.
1.2.1
The Lagrangian
In order to formulate Lagrangian mechanics we need a way to describe physical objects and their motions. We do this by giving a parameterisa-tion, which requires two main ingredients: a parameter space and a target space. The parameter space tells us what the object looks like and the time period we are interested in. For simplicity we will always assume our object is a smooth connected manifold M, which justifies us to take the pa-rameter space to be M×R. Here R represents the time domain. The target space is the space in which the object moves. In the simplest case this is flat spacetime, where the metric is constant, which is exactly Minkowski space. These observations justify the following definition.
Definition 1.7. A parameterisation is a differentiable map X : R×M → R(1,d)written as X = (X0, X1, . . . , Xd) = (X0,~X)such that the following hold:
• For each(t, m) ∈R×M the vector ∂
∂tX(t, m)is either timelike of lightlike.
The above requirements are clearly necessary for our definition to repre-sent any physical object, but they will not be sufficient for specific pur-poses. For instance, no physical laws are incorporated within this defini-tion, hence additional requirements will be made dependent on the spe-cific context. The way this will be done is through a Lagrangian.
We will formulate Lagrangian mechanics in natural units. This means we set
c =¯h =1, (1.19)
where c is the speed of light, and ¯h the reduced Planck constant. This means we measure time, distance and inverse energy in the same units. Especially the fact that we do not distinguish between distance and time units seems reasonable, since relativity tells us space and time are not fully distinct but can be mixed instead. The biggest advantage of natural units is that it cleans up formulas without loss of information, as the correct quantities of c and ¯h can always be recovered by comparing units.
Definition 1.8. A Lagrangian density is a map which assigns to each parma-terisation X a functionL(X) : R×M →R, which depends differentiably on X and the derivatives of X. The associated Lagrangian is the function L(X) given by
L(X) :=
Z
M
L(X) dV. The Lagrangian has units of energy.
Definition 1.9. Given a parameterisation X and a Lagrangian L as in defini-tion 1.8 and a closed interval [ti, tf] with more than one point, the action is
given by S[ti,tf](X) := Z tf ti Z ML(X) dV dt= Z tf ti L(X)dt. (1.20) Often the subscript will be omitted if the interval is clear.
It follows that the action is dimensionless in natural units. We are now able to state the primary axiom of Lagrangian mechanics.
Definition 1.10. A parameterisation X is said to describe a (classical) physical object if for every closed interval [ti, tf] and every differentiable map Y : R×
M→R(1,d) such that for all m∈ M, Y(m, t
i) = Y(m, tf) =0, we have
∂
1.2 Lagrangian mechanics 17
Solving this equation generally fixes a set of equations of motion, which may need to be supplemented with separate boundary conditions. It is important to note that although an action can be used to fully fix the equa-tions of moequa-tions, the action that does this is not necessarily unique. For example, scalar multiples of an action give the same equations of motion, but actions which look vastly different may also have the same solutions. This can be exploited when studying a system, as different actions can expose different properties, such as symmetries, of a system. This is some-thing we will use in the study of the conformal properties of the free string, inchapter 3.
Another useful result from Lagrangian mechanics states that if L(X)only depends on X and the first order derivatives of X, and if X explicitly de-pends on a set of parameters ζ1, ..., ζn, then the requirement of the action
being stationary in X is equivalent to the Euler-Lagrange equation, which is given by ∂L ∂X − n
∑
i=1 ∂ ∂ζi Pζi =0 (1.22)wherePζi are the conjugate momenta defined by
Pζi = ∂L
∂(∂X∂ζ
i)
. (1.23)
1.2.2
The free relativistic point particle
We will discuss a short example of Lagrangian mechanics, namely the free relativistic point particle. By definition 1.7 a parameterisation of a point particle is a map X : R× {point} → R(1,d), but to simplify notation we will view this as a map X : R →R(1,d)
. Given a time interval[ti, tf] ⊆ R,
the action is then given by the proper time that has elapsed on the path that our particle traces out in the target space. This means we set
L=m s − ∂X ∂t 2 (1.24) and consequently S(X) = m Z tf ti s − ∂X ∂t 2 dt. (1.25)
The mass is included to make sure the dimensions are correct. For simple parameterisations which are taken in the static guage, which means X(t) = (t,~X(t))for all t, this means we take
S(X) = m Z tf ti v u u t1− ∂ ~ X ∂t !2 dt. (1.26)
Note that since the particle is assumed to travel slower than the speed of light, ∂X
∂t is assumed to be timelike. Therefore, the term within the square
root is indeed positive, and this expression is well-defined. If we were now given a path Y as indefinition 1.10, we have
S(X+eY) =m Z tf ti s − ∂(X+eY) ∂t 2 dt. =m Z tf ti s − ∂X ∂t 2 −2e ∂X ∂t ∂Y ∂t −e2 ∂Y ∂t 2 dt. Here we can do a Taylor expansion with respect to e, and discard any terms of order 2 or higher. We will also introduce the notation ˙X = ∂X
∂t. This gives S(X+eY) ≈m Z tf ti q −(X˙)2− e ˙ X q −(X˙)2 ˙ Y dt.
We can now calculate ∂
∂eS(X+eY) directly, were we use integration by
parts in the third equality:
∂ ∂eS(X+eY) =elim→0 S(X+eY) −S(X) e =m Z tf ti ˙ X q −(X˙)2 ˙ Y dt =m ˙ X q −(X˙)2 Y tf ti −m Z tf ti d dt ˙ X q −(X˙)2 Y dt.
But by the assumptions made on Y in definition 1.10 we have Y(ti) =
1.3 The quantum formalism 19
has to vanish for every path Y, and this gives us
md dt ˙ X q −(X˙)2 =0. (1.27)
To interpret this result, we return to the static gauge, where X(t) = (t,~X(t)). Here we can define ~V(t) = dtd~X, which gives us ˙X = (1,~V(t)). Additionally √ 1
−(X˙)2 becomes 1
√
1−~V2, which can be recognised as the
rel-ativistic Lorentz factor γ. If we now look at the spacial part of (1.27), we find dtd(γV~) = 0, which tells us exactly that the relativistic momentum
~p=γm~V (1.28)
is conserved. This is the known equation of motion for a single particle in the absence of forces, which establishes the idea the Lagrangian was chosen properly. Moreover, this implies the direction of V is constant.~ Additionally, since we have
1 1+ (γ~V)2 = 1 1+ V~2 1−~V2 = 1− ~V 2 1− ~V2+ ~V2 =1− ~V 2, (1.29)
the conservation of γV implies the conservation of~ ~V2, and so ~V is itself constant. Therefore, a full description of the trajectory of the point parti-cle can now be described by fixing a value for ~V(0) = ~V0, from which it
follows that ~ X(t) = ~X(0) + Z t 0 ~ V(t0)dt0 = ~X(0) + Z t 0 ~ V0dt0 = ~X(0) +t~V0. (1.30)
Hence the full trajectory is determined by the initial position and the initial velocity, or equivalently by the initial position and the initial momentum.
1.3
The quantum formalism
The Lagrangian formalism described in the previous section can be used to describe a classical system. Given a Lagrangian, equations of motion can be derived, which tell us exactly which parameterisations are allowed. These equations of motion usually take the shape of a set of differential equations, which in general do not have a single unique solution, but more
so a family of solutions, characterised by some free variables. For example, the relativistic point particle can be completely described by fixing an ini-tial position~X0and an initial momentum~p0. In a quantum theory of this
classical system these free variables are now turned into Hermitian oper-ators on some state space, with appropriate commutation relations. Here, one can opt for either Schr ¨odinger or Heisenberg operators. Schr ¨odinger operators remain constant in time while their states change, while Heisen-berg operators change over time while their states remain the same. Since inchapter 3we will work in the Heisenberg picture, we will now illustrate how one can transition between the two pictures.
In the case where the Schr ¨odinger Hamiltonian is time-independent, any state ψ evolving according to the Schr ¨odinger equation
i∂
∂tψ(t) = Hψ(t) (1.31)
satisfies ψ(t) = e−iHtψ. Therefore, the operator eiHt sends any time
de-pendent state to a time indede-pendent one. Hence, given any Schr ¨odinger operatorO, the corresponding Heisenberg operator isO(t) = eiHtOe−iHt, since for any state ψ
O(t)ψ=eiHtOe−iHtψ=eiHt(Oψ(t)), (1.32)
which is indeedOψ(t) brought to rest. Consequently, the time evolution
of a Heisenberg operator can neatly be calculated from its commutator with the Hamiltonian, via
∂
∂tO(t) = ∂ ∂t(e
iHtOe−iHt)
= (iHeiHt)Oe−iHt+eiHtO(−iHe−iHt) =iHO(t) −iO(t)H
=i[H,O(t)].
(1.33)
We will now derive one final result, which states that for any two Schr ¨odinger operators O and U the equality [O(t),U (t)] = [O,U ](t) holds. This simply follows from the following calculation:
[O(t),U (t)] = O(t)U (t) − U (t)O(t)
=eiHtUe−iHteiHtOe−iHt−eiHtUe−iHteiHtOe−iHt =eiHtU Oe−iHt−eiHtU Oe−iHt
=eiHt[O,U ]e−iHt = [O,U ](t).
1.3 The quantum formalism 21
This result is for example useful in the common case when one has two Schr ¨odinger operators q and p, which represent some position and the corresponding conjugate momentum. These operators should have the canonical commutation relation [q, p] = i, hence it follows their Heisen-berg counterparts satisfy [q(t), p(t)] = i. More generally, (1.34) allows us to (often implicitly) switch between the Schr ¨odinger and Heisenberg pictures more easily. This will be an advantage in our discussion of the quantum theory of the bosonic string inchapter 3. First however, we treat the shorter example of the quantum harmonic oscillator.
1.3.1
The quantum harmonic oscillator
In order to establish a quantum theory of the harmonic oscillator, we first describe the classical theory. A classical harmonic oscillator is a physical system whose kinetic energy is proportional to its velocity squared, and its potential energy is proportional to its position squared. Hence, if the position of the particle is given by q(t), the Lagrangian can be written as L(t) = 12ζ ˙q(t)2−12ηq(t)2, for suitable constants ζ and η. Since scaling of L
does not change the behaviour of the system, we may assume it is of the form L= 1 2ω ˙q 2− ω 2q 2. (1.35)
Then the conjugate momentum to q is
p= ∂L
∂ ˙q =
1
ω ˙q. (1.36)
and the Hamiltonian is given by
H = p˙q−L = 1 2ω ˙q 2+ω 2q 2= ω 2(p 2+q2). (1.37)
In order to quantise this oscillator, we introduce Hermitian Schr ¨odinger operators p and q, with canonical commutation relation [q, p] = i. We then define the operators a = √1
2(p−iq) and its adjoint a † = √1
2(p+iq).
A direct calculation then tells us that [a, a†] = 1, and we can retrieve p and q as p = √1
2(a+a
†) and q = √i
2(a−a
Hamiltonian as H = ω 2(p 2+q2) = ω 4 (a+a†)2− (a−a†)2 = ω 4 (a2+a†2+aa†+a†a) − (a2+a†2−aa†−a†a) = ω 2 aa†+a†a= ω 2 aa†+aa†− [a, a†]=ω(aa†−1 2). (1.38)
Here we recognise a† and a as raising and lowering operators respectively, since for any eigenstate|hiof H with eigenvalue h, we have
H(a|hi) =ω(aa†−1 2)a|hi = a ω(a†a−1 2)|hi =a(H−ω)|hi) = (h−ω)(a|hi) (1.39) and H(a†|hi) =ω(aa†−1 2)a †|hi = ω(a†a+1 2)a †|hi =a†(H+ω)|hi) = (h+ω)(a†|hi). (1.40)
Hence a† and a raise and lower the H eigenvalue (or energy) of states, which justifies the name. They are also often called creation and annihilation operators, as they take this role in a more general quantum field theory. Additionally, (1.38) gives us
ωa(t)a†(t) = eiHtaa†e−iHt =eiHt(H+ ω
2)e
−iHt = H+ω
2 =ωaa
†. (1.41)
We can now compute ˙a(t)using all previous results:
∂ ∂ta(t) (1.33) = i[H, a(t)](1.38=) iω[aa†−1/2, a(t)] (1.41) = iω[a(t)a†(t), a(t)] =iωa(t)[a†(t), a(t)] (1.34) = iωa(t)[a†, a](t) = −iωa(t). (1.42)
The differential equation is solved by
a(t) = e−iωta, (1.43) and consequently a†(t) = eiωta†. (1.44) We then obtain q(t) = √i 2(ae −iωt−a†eiωt). (1.45)
1.3 The quantum formalism 23
1.3.2
Symmetries and Noether’s theorem
So far we have formally defined what we view as a classical or quantum system. In this thesis we are primarily interested in systems that have con-formal symmetry. The precise definition of a symmetry depends on the context. In our framework, where we have some Lagrangian L, a symme-try is a smooth map h : R(1,d) →R(1,d) such that L(h◦X) = L(X), and h is everywhere locally bijective, which means its derivative is always invert-ible. Symmetries can often be described by one-parameter families, which are sets {hs}s∈R of symmetries such that hs+t = hs ◦ht, and in particular
h0 = id. It is important to note such families are fully determined once
they are known in any neighbourhood of s = 0. Hence, a one-parameter family of symmetries is already fully fixed once one knows its behaviour around s =0, i.e. once one knows
δh:= ∂hs ∂s s=0 .
The map δh is called the infinitesimal transformation corresponding to the family{hs}s∈R. The generator of the transformations is the differential op-erator δh∇, where ∇ is some formal differential operator acting on the parameter space, whose precise definition depends on the specific con-text. It can be shown that the set of the differential operators obtained this way is closed under the commutator, and hence this set forms an algebraic structure which is known as a Lie algebra [BK13].
On the other hand, Noether’s theorem tells us that for such a family{hs}s∈R we get a conserved charge Q, which in the quantum theory becomes a Hermitian operator. This operator then generates the symmetry transfor-mation the classical charge originated from, and the set of these operators is also closed under the commutator, and we again obtain a Lie algebra. The two Lie algebras we have obtained are connected: the algebra of the quantum operators is isomorphic to (a central extension of) the complexifi-cation of the algebra of classical generators. For example, in the next chap-ter we will show that the classical generators of conformal transformations form the Witt algebra, which implies that the corresponding quantum the-ory should be a representation of the central extension of the Witt algebra, which is the Virasoro algebra.
A similar argument can be made for Lorentz invariance. The Lagrangian of the harmonic oscillator which we previously analysed is invariant un-der Lorentz transformations, and Noether’s theorem hence implies there
are conserved charges. Explicitly, these charges are given by
Mµν =xµpν−pνxµ, (1.46)
which in the quantum theory turn into operators with commutation rela-tions
[Mµν
, Mρσ] = iηµρMνσ−iηνρMµσ+iηµσMρν−iηνσMρµ
. (1.47) These commutation relations defines the Lorentz lie algebra. Any quan-tum theory which is compatible with special relativity has to be Lorentz invariant, and therefore it has to contain a set of operators satisfying these relations. In chapter 3, this fact will be used to conclude bosonic string theory can only exist in 26 spacetime dimensions.
In conclusion, we can describe symmetries of both classical and quantum systems through Lie algebras. This inspires us to study Lie algebras in more detail. In particular, we will in detail study the Lie algebra associated to conformal symmetries, namely the Virasoro algebra.
Chapter
2
The Virasoro algebra
In this chapter we study Lie algebras, the natural algebraic structure as-sociated to symmetries. In particular, we will study the Virasoro algebra, which is the Lie algebra associated to conformal transformations. We start with the general theory of algebras and Lie algebras, including the nec-essary concepts of homomorphisms and ideals. We then define represen-tations of Lie algebras and the universal enveloping algebra, and we state the Poincar´e-Birkhoff-Witt theorem. We conclude by deriving the Witt and Virasoro algebras as the Lie algebras corresponding to conformal transfor-mations.
2.1
Lie algebras
In this section,F is a field, and all vector spaces are taken over F. However, the less mathematically oriented reader can always think ofF as either R orC, as in physics we only encounter Lie algebras over these two fields. Before we define Lie algebras, we first define the more general concept of algebras.
Definition 2.1. An algebra is a vector space g equipped with a bilinear multi-plication[·,·]: g×g→g, which is called the bracket. If[·,·]is associative, then gis called an associative algebra. If there is an element 1∈ gsuch that for all X ∈ gwe have[1, X] = [X, 1] = X, then g is called unital and e is called a unit. As is common practice in algebra, we define the relevant morphisms be-tween algebras as the maps which in some way respect the algebraic struc-ture.
Definition 2.2. Let (g,[·,·]g),(h,[·,·]h) be algebras. An algebra
homomor-phism is a linear map φ : g→hsuch that for all X, Y ∈ g
φ([X, Y]g) = [φ(X), φ(Y)]h. (2.1)
If φ is a bijection, we call it an algebra isomorphism.
Just as in the case of group or ring theory, we will also look at quotient spaces and formulate an isomorphism theorem.
Definition 2.3. Let(g,[·,·])be an algebra. A linear subspace i⊆gis called • a subalgebra if[i, i] ⊆ i(i.e. for all E, F ∈iwe have[E, F] ∈i); • a (two-sided) ideal if[g, i] ⊆ iand[i, g] ⊆i.
The group theoretic analogues of subalgebras and ideals are subgroups and normal subgroups respectively. Accordingly, an ideal of a Lie algebra allow us to define the corresponding quotient space, which is also a Lie algebra.
Proposition 2.4. Let(g,[·,·])be an algebra, and i ⊆ gan ideal. Let q := g/i be the quotient vector space, and denote its elements asbXe := X+i. Then q is also an algebra with bracket defined by[bXe,bYe] = b[X, Y]e.
Proof. We only need to show the definition of this bracket is representative independent. So let X, X0, Y ∈ g, such that E=X−X0 ∈ i. Note[E, Y] ∈i, sob[E, Y]e = 0. Hence we have b[X, Y]e = [(X−X0) +X0, Y] =b[E, Y]e + [X0, Y] = [X0, Y] . (2.2) The proof for representative independence of the second coordinate is done in a similar fashion.
We now have all the necessary tools to formulate the first isomorphism theorem for Lie algebras.
Theorem 2.5. Let(g,[·,·]g)and(h,[·,·]h)be algebras, and φ : g→han algebra
homomorphism. Then: 1. ker(φ)is an ideal of g;
2. im(φ)is a subalgebra of h;
3. there exists a unique algebra isomorphism φ : g/ker(φ) → im(φ) such
that for all X∈ gwe have φ(g) =φ(bXe).
Proof. Note the first isomorphism theorem for vector spaces already tells us ker(φ)and im(φ)are subspaces, and that there exists a φ : g/ker(φ) →
im(φ)that is a vector space isomorphism. We therefore only need to proof
2.1 Lie algebras 27
1. For E ∈ ker(φ) and X ∈ g we have φ([E, X]g) = [φ(E), φ(X)]h =
[0, φ(X)]h = 0, so [E, X]g ∈ ker(φ). A similar argument gives
[X, E]g ∈ ker(φ), and hence ker(φ)is an ideal.
2. For Z, W ∈ im(φ), we may write Z= φ(X)and W =φ(Y)for some
X, Y ∈ g. Hence [Z, W]h = [φ(X), φ(Y)]h = φ([X, Y]g) ∈ im(φ), so
im(φ)is a subalgebra.
3. Let X, Y ∈ g. Then φ([bXe,bYe]g/i) = φ(b[X, Y]ge) = φ([X, Y]g) =
[φ(X), φ(Y)]h =
φ(bXe), φ(bYe)h. Thus φ is an algebra
homomor-phism, and since it’s an isomorphism of vector spaces, it is an algebra isomorphism.
We are specifically interested in a certain class of algebras, namely the Lie algebras.
Definition 2.6. A Lie algebra is an algebra(g,[·,·]), such that[·,·]is alternat-ing∗(i.e. [X, X] = 0 for all X ∈ g), and for all X, Y, Z ∈ gwe have the so called Jacobi identity
[X,[Y, Z]] + [Y,[Z, X]] + [Z,[X, Y]] =0. (2.3) The bracket of a Lie algebra is called a Lie bracket, and a subspace h ⊆gthat is closed under the Lie bracket is called a Lie subalgebra.
We first give a couple of examples of Lie algebras.
Example 2.7. (i) Take g = F3, and take the Lie bracket equal to the cross-product, which is defined as
x1 x2 x3 , y1 y2 y3 = x1 x2 x3 × y1 y2 y3 = x2y3−x3y2 x3y1−x1y3 x1y2−x2y1 . (2.4)
(ii) Let A be an associative algebra, and define [·,·] : A×A → A as the commutator: [X, Y] = XY−YX. Then (A,[·,·]) is a Lie algebra. A common example is to take A = EndF(V) with V a vector space, with composition as multiplication.
(iii) Let n ∈ N, and let V an n-dimensional vector space. Define sl(V) = {X ∈ EndF(V)|Tr(X) = 0} as the set of traceless endomorphisms of V, and define the Lie bracket on sl(V) again as the commutator. To see this is
∗Any bilinear map which is alternating is also antisymmetric. The converse is also
well-defined, let X, Y ∈ EndF(V), choose a basis of V and write X and Y in matrix form as X= (Xij)1≤i,j≤n, Y = (Yij)1≤i,j≤n. Then
Tr(XY) = n
∑
i=1 (XY)ii = n∑
i=1 n∑
j=1 XijYji = n∑
j=1 n∑
i=1 XijYji = n∑
i=1 (YX)i,i =Tr(YX). (2.5)So in particular, for any X, Y ∈ sl(V) we have Tr([X, Y]) = Tr(XY− YX) = Tr(XY) −Tr(YX) =Tr(XY) −Tr(XY) =0, so[X, Y] ∈ sl(V). Since in general for X, Y∈ sl(V)the product XY is not contained in sl(V), sl(V) is not an F algebra in the standard way (i.e. with composition as multiplication). Therefore, if one would want to study this space, it would be natural to view it as a Lie algebra.
2.2
Lie algebra representations and the universal
enveloping algebra
One should recall the main reason we study Lie algebras (in this thesis) is because they are the natural structure formed by symmetries. As out-lined insection 1.3.2, such a Lie algebra appears in the quantum theory as a set of operators on the state space. In mathematical terms, this means the state space being a representation of the Lie algebra. In this section we formally define Lie algebra representations, and we construct the univer-sal enveloping algebra: an associative algebra which can be used to further study representations.
We first give the definition of a Lie algebra representation.
Definition 2.8. Let(g,[·,·])be an Lie algebra. A Lie algebra representation of g is a pair(V, ρ), with V a vector space and ρ : g → EndF(V)a Lie algebra homomorphism. Here we view EndF(V)as a Lie algebra with the commutator as Lie bracket.
Although according to this definition the pair(V, ρ)is the representation, it is common to be sloppy and refer to either V or ρ as “the representation”. However, both should always be kept in mind. Furthermore, the sentence “Let(V, ρ)be a representation of g” should be understood as “Let V be a vector space and let ρ : g→EndF(V)be a Lie algebra homomorphism”. Since in general V and ρ can be very complicated, it is natural to look
2.2 Lie algebra representations and the universal enveloping algebra 29
at the “smallest” possible nontrivial representations. These are so called irreducible representations.
Definition 2.9. Let (g,[·,·]) be a Lie algebra, V a vector space, and ρ : g → EndF(V) a Lie algebra representation. Let W ⊆ V be a subspace. Now define
ρW : g → HomF(W, V) by X 7→ ρ(X)|W. If for all X ∈ gthe map ρW(X)
is an endomorphism of W, which means the image of ρW(X) is contained in W,
we may view ρW as a map from g to EndF(W). In this case, we call(W, ρW) a
subrepresentation of(V, ρ). The subrepresentations(V, ρ)and(0, 0)are called the trivial subrepresentations of(V, ρ). If(V, ρ)has exactly two subrepresen-tations, then it is called irreducible.
Since in general the image of ρ is not closed under composition, one may try to extend g to a larger structure on which ρ does have this property. The most obvious choice for this larger structure would be an unital associative algebra, since EndF(V)already has this structure. More concrete, we will embed g into an unital associative algebraU (g)with a map i : g ,→ U (g), such that every representation ρ : g → EndF(V) has a natural unique extension to aF-linear map ˜ρ : U (g) → EndF(V). This U (g) can be con-structed fairly easily, which we will do now.
Definition 2.10. Let (g,[·,·]) be a Lie algebra. For all n ∈ N, let g⊗n be the
tensor product of n copies of g (in particular g⊗0∼=F), and define T (g) := M
n∈N
g⊗n. (2.6)
We will now define a natural multiplication on T (g). Let p, q ∈ N, and let X = Np
i=1Xi ∈ g
⊗p and Y =Nq
j=1Yj ∈ g
⊗q. We then define a multiplication
· : g⊗p×g⊗q →g⊗(p+q) by: X·Y = p O i=1 Xi ! · q O j=1 Yj := p+q O k=1 Zk, (2.7) Zk = ( Xk for 1≤k ≤ p Yk−p for p<k≤ p+q. (2.8)
In other words, we concatenate X and Y. Note that for all n ∈ N, we may view g⊗n as a subset of T (g). This allows us to extend our newly defined multipli-cation· bilinearly toT (g). Since S
n∈Ng⊗n generatesLn∈Ng⊗n = T (g), we have now defined · on the entirety of T (g). Note · is associative, and has unit
1 ∈ g⊗0. This makes T (g) into a unital associative algebra, which we will call the tensor algebra of g. We now define S to be the ideal generated by the set {X⊗Y−Y⊗X− [X, Y] : X, Y ∈ g = g⊗1}, and we define the universal enveloping algebraU (g)as
U (g) := T (g)/S. (2.9) We write the elements ofU (g)as
$ p O i=1 Xi ' =: X1X2. . . Xp. (2.10)
This definition might seem very complicated and arbitrary at first glance, but the constructed algebraU (g) has exactly the universal property men-tioned before. This is summarised in the following theorem.
Theorem 2.11. Let (g,[·,·]) be a Lie algebra and i : g → U (g) the natural embedding. Then U (g) has the following universal property: for every unital associative algebra A and every Lie algebra homomorphism† φ : g → A there
exists a unique algebra homomorphism φ : U (g) → A which sends 1 ∈ U (g) to 1∈ A and satisfies φ◦i =φ.
Proof. Let A be an unital associative algebra, and let φ : g → A be a Lie algebra homomorphism. Let q : T (g) → U (g) denote the quotient map. First of all, we define j : g→ T (g)by X 7→ X ∈ g⊗1, where we view g⊗1 as a subset ofT (g). Note j is injective, and since no non-trivial element of g = j(g) is contained in S, so is i := q◦ j. Furthermore, since both q and j are linear, so is i. Also note g⊗1generatesT (g)in a multiplicative sense (allowing addition, multiplication and scalar multiplication). This allows us to define the natural extension of φ, namely ˜φ : T (g) → A which is
determined by the identity ˜ φ p O i=1 Xi ! = p
∏
i=1 φ(Xi), (2.11)and extended linearly. In particular it maps 1 ∈ T (g) to 1 ∈ A. It is easy to see that ˜φ◦j =φ. Now for any X, Y ∈ gwe have:
˜
φ(X⊗Y−Y⊗X− [X, Y])
=φ˜(X⊗Y) −φ˜(Y⊗X) −φ˜([X, Y])
=φ˜(X)φ˜(Y) −φ˜(Y)φX˜ ) − (φ˜(X)φ˜(Y) −φ˜(Y)φX˜ )) =0.
(2.12)
2.2 Lie algebra representations and the universal enveloping algebra 31 g T (g) U (g) A j q φ φ˜ φ i=q◦j
Figure 2.1:Commutative diagram of relevant maps.
Hence the set{X⊗Y−Y⊗X− [X, Y] : X, Y ∈ g = g⊗1} is contained in ker ˜φ, so S is as well. Hence, by the isomorphism theorem (theorem 2.5),
there exists a unique algebra homomorphism φ : U (g) → A satisfying
φ◦q =φ˜. Therefore we have
¯
φ◦i=φ¯◦q◦j =φ˜◦j=φ. (2.13)
Finally, to see the unicity of ¯φ, note ˜φis the unique map such that ˜φ◦j =φ,
as there is no choice for the behaviour of ˜φon g. Any ¯φ0satisfying ¯φ0◦i =φ
would give us ¯φ0◦q◦j = φ, and hence ˜φ = φ¯0◦q. ¯φ is the unique map
satisfying this last property, and is therefore unique entirely. We conclude ¯φboth exists and is unique.
The most important consequence oftheorem 2.11is the fact that any rep-resentation (V, ρ) of a Lie algebra (g,[·,·]) automatically yields a map ¯ρ : U (g) → End(V). This allows us to transition between g representa-tions andU (g)representations without any problems. We will now state a useful theorem concerning the universal enveloping algebra, and partially proof it. It is the Poincar´e-Birkhoff-Witt theorem, which gives an easy way to construct a basis for such an algebra.
Theorem 2.12. (The Poincar´e-Birkhoff-Witt theorem) Let(g,[·,·])be a Lie alge-bra, and let B= (bi)i∈I be a totally ordered basis of g (that is, B is totally ordered
by some total ordering). Then the set
S(B):={bn1bn2. . . bnm : m∈ N, n1, . . . , nm ∈ I, bn1 bn2 . . . bnm}
(2.14) forms a basis ofU (g).
Proof. We will show S(B) generates U (B). A full proof which also shows all elements of S(B) are also linearly independent can be found in [Hum12]. By definition ofT (g), the set
U(B) := ( m O i=1 bni : m∈N, and n1, . . . , nm ∈ I ) ⊆ T (g) (2.15) of unordered products of basis vectors generatesT (g). Hence, the images of these products under the quotient map generate U (g). It is therefore sufficient to write an element of U(B)as a linear combination of elements of S(B). So let m ∈ N, and let n1, . . . , nm ∈ I. We continue the proof by
induction on m.
• For m = 0 or m = 1 there is nothing to prove, as the corresponding empty product or single element is certainly ordered.
• Now assume the statement holds for m−1. That is, the class of ev-ery (not necessarily ordered) product in U(B)of length m−1 can be written in terms of the ordered products of S(B). Our task is now to write our arbitrary product b :=bn1bn2. . . bnm of length m as a linear
combination of elements of S(B). Note that if the Lie bracket on g would be the zero map, then this certainly would not be a problem. Indeed, because for all X, Y ∈ g we have X⊗Y−Y⊗X ∈ S (with S defined as above (2.9)). Hence, XY =YX ∈ U (g), soU (g)is com-mutative and we can simply swap around all bi until we are done.
In general, the Lie bracket is not zero and the process is more com-plicated, since swapping two adjacent terms results in a commutator term. But still, since we are dealing with finite products, we will only need a finite amount of swaps (of adjecent terms) to arrive at a fully ordered product. Let N denote the least amount of swaps we need. We continue with induction on N.
– For N =0 we do not need any swaps, so we are done.
– Assume any product of length m for which less than N swaps will suffice to order it can be written in terms of elements of S(B). Now we look back at the product b = bn1bn2. . . bnm we
started with. Since we can order this product by N swaps, there is an index j ∈ {1, 2, . . . , m−1} such that the product bn1. . . bnj+1bnj. . . bnm only requires N−1 swaps. Now note we
have b =bn1. . . bnjbnj+1. . . bnm =bn1. . . bnj−1 bnj+1bnj+ [bnj, bnj+1] bnj+2. . . bnm. (2.16)
2.2 Lie algebra representations and the universal enveloping algebra 33
Since [bnj, bnj+1] ∈ g, we can write it in terms of the basis B =
(bi)i∈I as
[bni, bni+1] =
∑
i∈Iλibi, (2.17)
with all λi ∈ F, all but finitely many equal to zero. Thus it
follows b=bn1. . . bnj−1 bnj+1bnj + [bnj, bnj+1] bnj+2. . . bnm =bn1. . . bnj−1 bnj+1bnj+
∑
i∈I λibi ! bnj+2. . . bnm =bn1. . . bnj−1bnj+1bnjbnj+2. . . bnm | {z }only N−1 swaps required
+
∑
i∈I λibn1. . . bnj−1bibnj+2. . . bnm | {z } only m−1 terms . (2.18) Now the first term only requires N −1 swaps to be fully or-dered, and can thus be written as a linear combination of ele-ments of S(B), by our second induction hypothesis. All other terms (the ones grouped within the summation) are products of only m−1 terms, as bnj and bnj+1 have been swapped out infavour of some bi. Hence all these terms can also be written in
terms of elements of S(B), and consequently b can as well. This concludes both our induction proofs, and therefore the proof as a whole.
To wind up this section, we will prove one more lemma from represen-tation theory of Lie algebras that will come into use later. It is known as Schur’s lemma ([Hum12]).
Lemma 2.13. (Schur’s lemma) Let (g,[·,·]) be a Lie algebra, and let (V, ρ),(W, σ) be irreducible representations of g. Let φ : V → W be a homo-morphism of representations, that is, φ is a linear map such that for all X ∈ g and for all v ∈ V we have φ(ρ(X)(v)) = σ(X)(φ(v)). Then the following
statements hold:
1. Either φ = 0 or φ is an isomorphism of representations (that is, it is a bijective representation homomorphism).
2. If(V, ρ) = (W, σ), and if φ (which is now an endomorphism) has at least one eigenvalue λ∈ F, then ρ(φ) =λId.
Proof. 1. The most important point for our proof is the fact that both V and W are assumed to be irreducible representations. This is impor-tant, since both the image and the kernel of φ are in fact subrepresen-tations. To see this, let v∈ ker φ. Then for all X ∈g:
φ(ρ(X)(v)) = σ(X)(φ(v)) = σ(X)(0) = 0 (2.19)
Hence ρ(X)(v) ∈ ker φ. This means the linear subspace ker φ is closed under the action of g, and it is therefore a subrepresentation. Since V is assumed to be irreducible it is therefore either equal to V or to 0, and φ is therefore either 0 or injective.
In a similar fashion, any element w in the image of φ can be written as w=φ(v)for some v ∈V. Now for X ∈ g:
σ(X)(w) = σ(X)(φ(v)) =φ(ρ(X)(v)) (2.20)
And hence σ(X)(w) ∈ im φ, and we can again conclude im φ is a subrepresentation. Since W is irreducible, this means im φ is either equal to 0 or to W, and hence φ is either 0 or surjective.
Note that if φ is zero, it cannot be injective or surjective, since this would imply either V =0 or W = 0, but this is ruled out by the fact V and W are irreducible (the zero space is not irreducible since it has only one subrepresentation). Hence, φ is either zero or both injective and surjective, and thus an isomorphism.
2. Assume (V, ρ) = (W, σ), and assume φ has at least one eigenvalue
λ. From part 1 of this lemma, φ is either zero or a bijection. If φ is
the zero map, it is a multiple of the identity, so here there is nothing to prove. Now let v ∈ V be an eigenvector of φ with eigenvalue λ. Then for all X∈ gwe have:
φ(ρ(X)(v)) =ρ(X)(φ(v)) =ρ(X)(λv) = λρ(X)(v) (2.21)
Hence ρ(X)(v)is also an eigenvector of φ, with the same eigenvalue. In other words, the eigenspace of φ associated with λ is closed under the action of g, and it is therefore a subrepresentation of V! Since we assumed φ is not the zero map, this means this eigenspace is whole of V, which directly proves φ=λId.
It is worth noting that the condition that φ has at least one eigenvalue is automatically satisfied in the case that V is finite dimensional and F is
2.3 The Witt and Virasoro algebras 35
algebraically closed. This is because if V is finite dimensional, then φ has a characteristic polynomial. The fact that F is algebraically closed then immediately implies this characteristic polynomial has a solution, and this solution is exactly an eigenvalue.
Remark. Just like in the case of group representations, the notation is of-ten shorof-tened. When working with a certain fixed representation, the ρ is suppressed and for v ∈ V we simply write Xv instead of ρ(X)(v), as if X acted directly on V.
This finishes our discussion on Lie algebras in general. From here on out, we will focus on the Lie algebras corresponding to conformal symmetries.
2.3
The Witt and Virasoro algebras
To find the Lie algebra of conformal transformations, we have to find the infinitesimal generators of such transformations. Just as the quantum Hamiltonian generates time translation through eiaH|φ(t)i = |φ(t+a)i,
the infinitesimal generator T of some family{hs}of conformal
transforma-tions should satisfy esTf(x) = f(hs(x)). Recall conformal transformations
are exactly the holomorphic and anti-holomorphic maps on C. We now focus on the holomorphic ones. A holomorphic map can always locally be expressed as a (convergent) Laurent series in z, which can be approx-imated arbitrarily well by a polynomial p(z, z−1) as z 7→ z+ p(z, z−1). We therefore look for the generators of transformations of the form z 7→ z+azn(with n ∈ Z), as these transformations in turn generate such poly-nomials. We claim the generators are simply given by ln := −zn+1∂z.
Namely, for some smooth test function f with a locally convergent Taylor series, we have f(z−azn+1) = ∞
∑
k=0 1 k!f (k)( z)(−azn+1)k = ∞∑
k=0 (−azn+1∂z)k k! f(z) =exp(−azn+1∂z)f(z) =exp(aln)f(z).This shows the ln indeed generate holomorphic transformations.
n, m∈ Z and a test function f we namely have [`n,`m]f = (`n`m− `m`n)f =zn+1 ∂ ∂z zm+1∂ f ∂z −zm+1 ∂ ∂z zn+1∂ f ∂z =zn+1 (m+1)zm∂ f ∂z +z m+1∂2f ∂z2 −zm+1 (n+1)zm∂ f ∂z +z n+1∂2f ∂z2 = ((m+1) − (n+1))zn+m+1∂ f ∂z +z n+m+2∂2f ∂z2 −z n+m+2∂2f ∂z2 = −(n−m)zn+m+1∂ f ∂z = (n−m)`n+mf , (2.22) and hence[ln, lm] = (n−m)ln+m. This shows the set of infinitesimal
gen-erators is indeed closed under the commutator. Since the commutator is also clearly alternating and since it obeys the Jacobi identity (commutators defined by associative products in general do), it forms a Lie algebra. The complexification‡of this Lie algebra is known as the Witt algebra.
Definition 2.14. The Witt algebra is the complex Lie algebra whose vector space Wis generated by the set{`n}n∈Z, with Lie bracket given by
[ln, lm] = (n−m)ln+m.
Similarly, from the antiholomorphic conformal transformations we get an-other set of infinitesimal generators, which are given by ¯ln := −¯zn+1∂¯z.
These generators have the same commutation relations as the ln, and
therefore also form a copy of the Witt algebra. However, since these two copies commute with each other and thereby act independently, it is often sufficient to study only the holomorphic part.
When transitioning to a quantum theory, it is not the Witt algebra but a central extension of the Witt algebra that is encountered. We first state what we mean by a central extension [Sch08].
Definition 2.15. Let g be a Lie algebra. A Lie algebra extension of g by a Lie algebra c is a Lie algebra h with a short exact sequence 0 → c ,→i h p g → 0,
‡Formally, the complexification of a real Lie algebra R is the complex Lie algebra
whose vector space is given byC⊗Rg, with Lie bracket[λ⊗X, µ⊗Y] = λµ⊗ [X, Y].
If one is given a basis of g, the complexification can simply be understood as allowing complex coefficients instead of only real ones.
2.3 The Witt and Virasoro algebras 37
with i and p Lie algebra homomorphisms. If i(c)is central in h, i.e. for all X ∈h and C ∈ i(c)we have[X, C] =0, then h is called a central extension of g by c. If h∼=c⊕gas Lie algebras, then this extension is called trivial.
To gain some intuition as to why we would look at a central extension, assume we have some set of symmetries G of a quantum system, which is represented by a Hilbert spaceH. One might think this means there is some unitary action of G onH. However, the relevant objects within this Hilbert space are not simply its elements, but more so the rays ofH, which are one dimensional subspaces of H and by definition precisely the ele-ments of the projective spaceP(H). This space comes with an additional structure, namely a way to calculate transition probabilities. Given two rays|φi,|φitheir transition probability namely is given by
chance(|φi → |ψi) = p |hφ|ψi| hφ|φihψ|ψi
, (2.23)
which is indeed a well-defined operation on P(H). If we want G to act as a set of symmetries onP(H), its elements should respect this transition probability, so each f ∈ G acts as a map f : P(H) → P(H) such that for all|φi,|φiwe have
chance(f |φi → f |ψi) =chance(|φi → |ψi). (2.24)
By Wigner’s theorem ([Bar64]), such a map has a lift to a map ˆf :H →H, which is unique up to multiplication with some complex phase factor. This lift has to behave nicely under composition, which means the following: Given two general f , f0 acting onP(H), we can either take the composi-tion f ◦ f0, or we can choose to first lift them to ˆf, ˆf0, then take the com-position ˆf◦ ˆf0of these two and then let this new map act onP(H). These two maps onP(H)(which are f ◦ f0and the projection of ˆf◦ ˆf0) should be identical, so f[◦ f0and ˆf◦ ˆf0can at most differ by a phase factor. The effect of this on the symmetry Lie algebra is that we can lift the Lie algebra of in-finitesimal transformations g to a slightly bigger algebra ˆg :=g⊕C, which satisfies [X, ˆˆ Y] = \[X, Y] +λ(X, Y) for all X, Y ∈ g, where λ(X, Y) ∈ C
[Wened]. Hence the sequence
0→C→ ˆg→g→0 (2.25)
is exact. To summarise, if a quantum system has a set of infinitesimal sym-metries g, then these symsym-metries form a projective representation, which is equivalent to an ordinary representation of a central extension ˆg of g. This then finally brings us to the Lie algebra which takes the central role in this thesis; the Virasoro algebra.
Definition 2.16. The Virasoro algebra with central charge c∈ C∗is the Lie algebra whose vector space is given by
Vc :=SpanC({Ln}n∈Z∪ {I}), where the Lie bracket is defined by
[Ln, Lm] = (n−m)Ln+m− c
12(m
3−m)
δn+m,0I and [Ln, I] = 0, (2.26)
and extended linearly.
Note we have not shown (2.26) indeed defines a Lie algebra, and it is a priori not obvious it satisfies the Jacobi identity. The calculation however is more tedious than interesting, so we will not perform it here. Moreover, although we have defined a Virasoro algebra for every c∈ C∗, they are in fact all isomorphic. This can be seen by explicitly defining an isomorphism
φc : Vc →V1by
φc(Ln) = Ln and φc(I) = I
c,
and extending linearly. φc is clearly linear and bijective. Moreover, if we
let[·,·]c denote the bracket on Vc, for n, m ∈Z we have
φc([Ln, Lm]c) = φc (n−m)Ln+m− c 12(m 3−m) δn+m,0I = (n−m)Ln+m− c 12(m 3−m) δn+m,0φc(I) = (n−m)Ln+m− 1 12(m 3−m) δn+m,0I =φc([Ln, Lm]1).
Hence φc is a Lie algebra isomorphism. One could argue there is no need
to include a central charge in the definition of the Virasoro algebra, and simply always work with V1 instead. We will however not do this, as
in representations of the Virasoro algebra it is customary to let I act as the identity. The central charge allows us to without loss of generality to always assume this is the case.
As promised, the Virasoro algebra is a central extension of the Witt alge-bra, which we will now prove. Furthermore, it can be shown it is up to isomorphism, the only nontrivial central extension of the Witt algebra. Proposition 2.17. For every c ∈ C, the Virasoro algebra with central charge c is a nontrivial central extension of the Witt algebra, and is up to isomorphism the only one.
